We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^3)). Strict Trs: { terms(N) -> cons(recip(sqr(N))) , sqr(0()) -> 0() , sqr(s(X)) -> s(add(sqr(X), dbl(X))) , add(0(), X) -> X , add(s(X), Y) -> s(add(X, Y)) , dbl(0()) -> 0() , dbl(s(X)) -> s(s(dbl(X))) , first(0(), X) -> nil() , first(s(X), cons(Y)) -> cons(Y) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^3)) We add the following dependency tuples: Strict DPs: { terms^#(N) -> c_1(sqr^#(N)) , sqr^#(0()) -> c_2() , sqr^#(s(X)) -> c_3(add^#(sqr(X), dbl(X)), sqr^#(X), dbl^#(X)) , add^#(0(), X) -> c_4() , add^#(s(X), Y) -> c_5(add^#(X, Y)) , dbl^#(0()) -> c_6() , dbl^#(s(X)) -> c_7(dbl^#(X)) , first^#(0(), X) -> c_8() , first^#(s(X), cons(Y)) -> c_9() } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^3)). Strict DPs: { terms^#(N) -> c_1(sqr^#(N)) , sqr^#(0()) -> c_2() , sqr^#(s(X)) -> c_3(add^#(sqr(X), dbl(X)), sqr^#(X), dbl^#(X)) , add^#(0(), X) -> c_4() , add^#(s(X), Y) -> c_5(add^#(X, Y)) , dbl^#(0()) -> c_6() , dbl^#(s(X)) -> c_7(dbl^#(X)) , first^#(0(), X) -> c_8() , first^#(s(X), cons(Y)) -> c_9() } Weak Trs: { terms(N) -> cons(recip(sqr(N))) , sqr(0()) -> 0() , sqr(s(X)) -> s(add(sqr(X), dbl(X))) , add(0(), X) -> X , add(s(X), Y) -> s(add(X, Y)) , dbl(0()) -> 0() , dbl(s(X)) -> s(s(dbl(X))) , first(0(), X) -> nil() , first(s(X), cons(Y)) -> cons(Y) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^3)) We estimate the number of application of {2,4,6,8,9} by applications of Pre({2,4,6,8,9}) = {1,3,5,7}. Here rules are labeled as follows: DPs: { 1: terms^#(N) -> c_1(sqr^#(N)) , 2: sqr^#(0()) -> c_2() , 3: sqr^#(s(X)) -> c_3(add^#(sqr(X), dbl(X)), sqr^#(X), dbl^#(X)) , 4: add^#(0(), X) -> c_4() , 5: add^#(s(X), Y) -> c_5(add^#(X, Y)) , 6: dbl^#(0()) -> c_6() , 7: dbl^#(s(X)) -> c_7(dbl^#(X)) , 8: first^#(0(), X) -> c_8() , 9: first^#(s(X), cons(Y)) -> c_9() } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^3)). Strict DPs: { terms^#(N) -> c_1(sqr^#(N)) , sqr^#(s(X)) -> c_3(add^#(sqr(X), dbl(X)), sqr^#(X), dbl^#(X)) , add^#(s(X), Y) -> c_5(add^#(X, Y)) , dbl^#(s(X)) -> c_7(dbl^#(X)) } Weak DPs: { sqr^#(0()) -> c_2() , add^#(0(), X) -> c_4() , dbl^#(0()) -> c_6() , first^#(0(), X) -> c_8() , first^#(s(X), cons(Y)) -> c_9() } Weak Trs: { terms(N) -> cons(recip(sqr(N))) , sqr(0()) -> 0() , sqr(s(X)) -> s(add(sqr(X), dbl(X))) , add(0(), X) -> X , add(s(X), Y) -> s(add(X, Y)) , dbl(0()) -> 0() , dbl(s(X)) -> s(s(dbl(X))) , first(0(), X) -> nil() , first(s(X), cons(Y)) -> cons(Y) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^3)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { sqr^#(0()) -> c_2() , add^#(0(), X) -> c_4() , dbl^#(0()) -> c_6() , first^#(0(), X) -> c_8() , first^#(s(X), cons(Y)) -> c_9() } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^3)). Strict DPs: { terms^#(N) -> c_1(sqr^#(N)) , sqr^#(s(X)) -> c_3(add^#(sqr(X), dbl(X)), sqr^#(X), dbl^#(X)) , add^#(s(X), Y) -> c_5(add^#(X, Y)) , dbl^#(s(X)) -> c_7(dbl^#(X)) } Weak Trs: { terms(N) -> cons(recip(sqr(N))) , sqr(0()) -> 0() , sqr(s(X)) -> s(add(sqr(X), dbl(X))) , add(0(), X) -> X , add(s(X), Y) -> s(add(X, Y)) , dbl(0()) -> 0() , dbl(s(X)) -> s(s(dbl(X))) , first(0(), X) -> nil() , first(s(X), cons(Y)) -> cons(Y) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^3)) Consider the dependency graph 1: terms^#(N) -> c_1(sqr^#(N)) -->_1 sqr^#(s(X)) -> c_3(add^#(sqr(X), dbl(X)), sqr^#(X), dbl^#(X)) :2 2: sqr^#(s(X)) -> c_3(add^#(sqr(X), dbl(X)), sqr^#(X), dbl^#(X)) -->_3 dbl^#(s(X)) -> c_7(dbl^#(X)) :4 -->_1 add^#(s(X), Y) -> c_5(add^#(X, Y)) :3 -->_2 sqr^#(s(X)) -> c_3(add^#(sqr(X), dbl(X)), sqr^#(X), dbl^#(X)) :2 3: add^#(s(X), Y) -> c_5(add^#(X, Y)) -->_1 add^#(s(X), Y) -> c_5(add^#(X, Y)) :3 4: dbl^#(s(X)) -> c_7(dbl^#(X)) -->_1 dbl^#(s(X)) -> c_7(dbl^#(X)) :4 Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts). { terms^#(N) -> c_1(sqr^#(N)) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^3)). Strict DPs: { sqr^#(s(X)) -> c_3(add^#(sqr(X), dbl(X)), sqr^#(X), dbl^#(X)) , add^#(s(X), Y) -> c_5(add^#(X, Y)) , dbl^#(s(X)) -> c_7(dbl^#(X)) } Weak Trs: { terms(N) -> cons(recip(sqr(N))) , sqr(0()) -> 0() , sqr(s(X)) -> s(add(sqr(X), dbl(X))) , add(0(), X) -> X , add(s(X), Y) -> s(add(X, Y)) , dbl(0()) -> 0() , dbl(s(X)) -> s(s(dbl(X))) , first(0(), X) -> nil() , first(s(X), cons(Y)) -> cons(Y) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^3)) We replace rewrite rules by usable rules: Weak Usable Rules: { sqr(0()) -> 0() , sqr(s(X)) -> s(add(sqr(X), dbl(X))) , add(0(), X) -> X , add(s(X), Y) -> s(add(X, Y)) , dbl(0()) -> 0() , dbl(s(X)) -> s(s(dbl(X))) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^3)). Strict DPs: { sqr^#(s(X)) -> c_3(add^#(sqr(X), dbl(X)), sqr^#(X), dbl^#(X)) , add^#(s(X), Y) -> c_5(add^#(X, Y)) , dbl^#(s(X)) -> c_7(dbl^#(X)) } Weak Trs: { sqr(0()) -> 0() , sqr(s(X)) -> s(add(sqr(X), dbl(X))) , add(0(), X) -> X , add(s(X), Y) -> s(add(X, Y)) , dbl(0()) -> 0() , dbl(s(X)) -> s(s(dbl(X))) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^3)) We decompose the input problem according to the dependency graph into the upper component { sqr^#(s(X)) -> c_3(add^#(sqr(X), dbl(X)), sqr^#(X), dbl^#(X)) } and lower component { add^#(s(X), Y) -> c_5(add^#(X, Y)) , dbl^#(s(X)) -> c_7(dbl^#(X)) } Further, following extension rules are added to the lower component. { sqr^#(s(X)) -> sqr^#(X) , sqr^#(s(X)) -> add^#(sqr(X), dbl(X)) , sqr^#(s(X)) -> dbl^#(X) } TcT solves the upper component with certificate YES(O(1),O(n^1)). Sub-proof: ---------- We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { sqr^#(s(X)) -> c_3(add^#(sqr(X), dbl(X)), sqr^#(X), dbl^#(X)) } Weak Trs: { sqr(0()) -> 0() , sqr(s(X)) -> s(add(sqr(X), dbl(X))) , add(0(), X) -> X , add(s(X), Y) -> s(add(X, Y)) , dbl(0()) -> 0() , dbl(s(X)) -> s(s(dbl(X))) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We use the processor 'Small Polynomial Path Order (PS,1-bounded)' to orient following rules strictly. DPs: { 1: sqr^#(s(X)) -> c_3(add^#(sqr(X), dbl(X)), sqr^#(X), dbl^#(X)) } Sub-proof: ---------- The input was oriented with the instance of 'Small Polynomial Path Order (PS,1-bounded)' as induced by the safe mapping safe(sqr) = {}, safe(0) = {}, safe(s) = {1}, safe(add) = {1, 2}, safe(dbl) = {}, safe(sqr^#) = {}, safe(c_3) = {}, safe(add^#) = {}, safe(dbl^#) = {} and precedence empty . Following symbols are considered recursive: {sqr^#} The recursion depth is 1. Further, following argument filtering is employed: pi(sqr) = [], pi(0) = [], pi(s) = [1], pi(add) = [1, 2], pi(dbl) = 1, pi(sqr^#) = [1], pi(c_3) = [1, 2, 3], pi(add^#) = [], pi(dbl^#) = [] Usable defined function symbols are a subset of: {sqr^#, add^#, dbl^#} For your convenience, here are the satisfied ordering constraints: pi(sqr^#(s(X))) = sqr^#(s(; X);) > c_3(add^#(), sqr^#(X;), dbl^#();) = pi(c_3(add^#(sqr(X), dbl(X)), sqr^#(X), dbl^#(X))) The strictly oriented rules are moved into the weak component. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Weak DPs: { sqr^#(s(X)) -> c_3(add^#(sqr(X), dbl(X)), sqr^#(X), dbl^#(X)) } Weak Trs: { sqr(0()) -> 0() , sqr(s(X)) -> s(add(sqr(X), dbl(X))) , add(0(), X) -> X , add(s(X), Y) -> s(add(X, Y)) , dbl(0()) -> 0() , dbl(s(X)) -> s(s(dbl(X))) } Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { sqr^#(s(X)) -> c_3(add^#(sqr(X), dbl(X)), sqr^#(X), dbl^#(X)) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Weak Trs: { sqr(0()) -> 0() , sqr(s(X)) -> s(add(sqr(X), dbl(X))) , add(0(), X) -> X , add(s(X), Y) -> s(add(X, Y)) , dbl(0()) -> 0() , dbl(s(X)) -> s(s(dbl(X))) } Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) No rule is usable, rules are removed from the input problem. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Rules: Empty Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) Empty rules are trivially bounded We return to the main proof. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict DPs: { add^#(s(X), Y) -> c_5(add^#(X, Y)) , dbl^#(s(X)) -> c_7(dbl^#(X)) } Weak DPs: { sqr^#(s(X)) -> sqr^#(X) , sqr^#(s(X)) -> add^#(sqr(X), dbl(X)) , sqr^#(s(X)) -> dbl^#(X) } Weak Trs: { sqr(0()) -> 0() , sqr(s(X)) -> s(add(sqr(X), dbl(X))) , add(0(), X) -> X , add(s(X), Y) -> s(add(X, Y)) , dbl(0()) -> 0() , dbl(s(X)) -> s(s(dbl(X))) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. DPs: { 2: dbl^#(s(X)) -> c_7(dbl^#(X)) , 3: sqr^#(s(X)) -> sqr^#(X) , 4: sqr^#(s(X)) -> add^#(sqr(X), dbl(X)) , 5: sqr^#(s(X)) -> dbl^#(X) } Trs: { dbl(0()) -> 0() } Sub-proof: ---------- The following argument positions are usable: Uargs(c_5) = {1}, Uargs(c_7) = {1} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [sqr](x1) = [0] [0] = [0] [s](x1) = [1] x1 + [4] [add](x1, x2) = [4] x2 + [0] [dbl](x1) = [1] [sqr^#](x1) = [2] x1 + [0] [add^#](x1, x2) = [1] [dbl^#](x1) = [1] x1 + [6] [c_5](x1) = [1] x1 + [0] [c_7](x1) = [1] x1 + [0] The order satisfies the following ordering constraints: [sqr(0())] = [0] >= [0] = [0()] [sqr(s(X))] = [0] ? [8] = [s(add(sqr(X), dbl(X)))] [add(0(), X)] = [4] X + [0] >= [1] X + [0] = [X] [add(s(X), Y)] = [4] Y + [0] ? [4] Y + [4] = [s(add(X, Y))] [dbl(0())] = [1] > [0] = [0()] [dbl(s(X))] = [1] ? [9] = [s(s(dbl(X)))] [sqr^#(s(X))] = [2] X + [8] > [2] X + [0] = [sqr^#(X)] [sqr^#(s(X))] = [2] X + [8] > [1] = [add^#(sqr(X), dbl(X))] [sqr^#(s(X))] = [2] X + [8] > [1] X + [6] = [dbl^#(X)] [add^#(s(X), Y)] = [1] >= [1] = [c_5(add^#(X, Y))] [dbl^#(s(X))] = [1] X + [10] > [1] X + [6] = [c_7(dbl^#(X))] The strictly oriented rules are moved into the weak component. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict DPs: { add^#(s(X), Y) -> c_5(add^#(X, Y)) } Weak DPs: { sqr^#(s(X)) -> sqr^#(X) , sqr^#(s(X)) -> add^#(sqr(X), dbl(X)) , sqr^#(s(X)) -> dbl^#(X) , dbl^#(s(X)) -> c_7(dbl^#(X)) } Weak Trs: { sqr(0()) -> 0() , sqr(s(X)) -> s(add(sqr(X), dbl(X))) , add(0(), X) -> X , add(s(X), Y) -> s(add(X, Y)) , dbl(0()) -> 0() , dbl(s(X)) -> s(s(dbl(X))) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { sqr^#(s(X)) -> dbl^#(X) , dbl^#(s(X)) -> c_7(dbl^#(X)) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict DPs: { add^#(s(X), Y) -> c_5(add^#(X, Y)) } Weak DPs: { sqr^#(s(X)) -> sqr^#(X) , sqr^#(s(X)) -> add^#(sqr(X), dbl(X)) } Weak Trs: { sqr(0()) -> 0() , sqr(s(X)) -> s(add(sqr(X), dbl(X))) , add(0(), X) -> X , add(s(X), Y) -> s(add(X, Y)) , dbl(0()) -> 0() , dbl(s(X)) -> s(s(dbl(X))) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We use the processor 'custom shape polynomial interpretation' to orient following rules strictly. DPs: { 1: add^#(s(X), Y) -> c_5(add^#(X, Y)) , 2: sqr^#(s(X)) -> sqr^#(X) , 3: sqr^#(s(X)) -> add^#(sqr(X), dbl(X)) } Trs: { sqr(0()) -> 0() , dbl(0()) -> 0() } Sub-proof: ---------- The following argument positions are considered usable: Uargs(c_5) = {1} TcT has computed the following constructor-restricted polynomial interpretation. [sqr](x1) = 1 + x1^2 [0]() = 0 [s](x1) = 2 + x1 [add](x1, x2) = x1 + x2 [dbl](x1) = 2 + 2*x1 [sqr^#](x1) = 3 + x1^2 [add^#](x1, x2) = 2 + x1 [dbl^#](x1) = 0 [c_5](x1) = x1 [c_7](x1) = 0 This order satisfies the following ordering constraints. [sqr(0())] = 1 > = [0()] [sqr(s(X))] = 5 + 4*X + X^2 >= 5 + X^2 + 2*X = [s(add(sqr(X), dbl(X)))] [add(0(), X)] = X >= X = [X] [add(s(X), Y)] = 2 + X + Y >= 2 + X + Y = [s(add(X, Y))] [dbl(0())] = 2 > = [0()] [dbl(s(X))] = 6 + 2*X >= 6 + 2*X = [s(s(dbl(X)))] [sqr^#(s(X))] = 7 + 4*X + X^2 > 3 + X^2 = [sqr^#(X)] [sqr^#(s(X))] = 7 + 4*X + X^2 > 3 + X^2 = [add^#(sqr(X), dbl(X))] [add^#(s(X), Y)] = 4 + X > 2 + X = [c_5(add^#(X, Y))] The strictly oriented rules are moved into the weak component. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Weak DPs: { sqr^#(s(X)) -> sqr^#(X) , sqr^#(s(X)) -> add^#(sqr(X), dbl(X)) , add^#(s(X), Y) -> c_5(add^#(X, Y)) } Weak Trs: { sqr(0()) -> 0() , sqr(s(X)) -> s(add(sqr(X), dbl(X))) , add(0(), X) -> X , add(s(X), Y) -> s(add(X, Y)) , dbl(0()) -> 0() , dbl(s(X)) -> s(s(dbl(X))) } Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { sqr^#(s(X)) -> sqr^#(X) , sqr^#(s(X)) -> add^#(sqr(X), dbl(X)) , add^#(s(X), Y) -> c_5(add^#(X, Y)) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Weak Trs: { sqr(0()) -> 0() , sqr(s(X)) -> s(add(sqr(X), dbl(X))) , add(0(), X) -> X , add(s(X), Y) -> s(add(X, Y)) , dbl(0()) -> 0() , dbl(s(X)) -> s(s(dbl(X))) } Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) No rule is usable, rules are removed from the input problem. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Rules: Empty Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) Empty rules are trivially bounded Hurray, we answered YES(O(1),O(n^3))